The Ehrenfest Paradox is the single (arguably) remaining unresolved paradox in Einstein’s Special Relativity Theory. It proposes that relativistic Lorentz contraction around the circumference of a rotating disc or cylinder will alter the value of π.
Einstein’s 1905 Special Relativity Theory (SRT) holds that an observer in any sort of inertial motion will measure a fixed speed of light, ‘c’. This immediately produces three effects for observers in different inertial reference frames.
- Time will appear slowed down in the other frame – time dilation.
- Mass in the other frame will appear increased.
- Length in the other frame will appear shortened in the direction of travel – Lorentz contraction.
Further examination of momentum and energy produces Einstein’s signature equation, E=mc˛.
Both time dilation and mass increase are observed in particle accelerators. Lorentz contraction has yet to be demonstrated, since anything large enough to have a measurable length at relativistic speeds is too heavy to accelerate that fast. Mathematically, though, it follows from the two observed effects.
In 1909, Paul Ehrenfest proposed a paradox that has physicists tied in knots to this day. Einstein found one solution that led him to consider non-Euclidean geometry, curved space, which led him on to General Relativity (GRT) in 1916.
Briefly, then, here is the paradox –
If the circumference of a rotating disc travels at a relativistic speed, the circumferential length should contract in the direction of travel. The radius of the disc travels sideways, not in the direction of its length, so its length should be unchanged. π (Greek letter pi) is the circumference divided by twice the radius, but circumference here has contracted, therefore the value of π has decreased, and varies with the rim speed.
That was Einstein’s conclusion, but here’s another take on the paradox: Measuring rods placed around the moving rim will contract, requiring more of them to go all the way around. Thus the circumference now increases, and the value of π also increases.
Opposite results, equally viable. A cynical take would be that the contraction of the rods equals the contraction of the rim, therefore the value of π remains the same.
We’ll take a different approach here, looking at the nature of contraction itself.